The Negative Binomial INAR(1) Process under Different Thinning Processes: Can We Separate between the Different Models?

The literature on discrete valued time series is expanding very fast. Very often we see new models with very similar properties to the existing ones. A natural question that arises is whether the multitude of models with very similar properties can really have a practical purpose or if they mostly present theoretical interest. In the present paper, we consider four models that have negative binomial marginal distributions and are autoregressive in order 1 behavior, but they have a very different generating mechanism. Then we try to answer the question whether we can distinguish between them with real data. Extensive simulations show that while the differences are small, we still can discriminate between the models with relatively moderate sample sizes. However, the mean forecasts are expected to be almost identical for all models.